1. Field of the Invention
The present invention generally relates to methods, systems and software product used in the area of computer-aided engineering analysis (e.g., finite element method (FEM), meshfree method, etc.), more particularly to methods and systems for numerically simulation structural behaviors of embedded bi-materials (e.g., particle-reinforced composites, fiber-reinforced composites, etc.).
2. Description of the Related Art
A composite material is a microscopic or macroscopic combination of two or more distinct materials with a recognizable interface between them. Composite materials are developed because no single, homogeneous structural material could be found that had all of the desired characteristics for a given application. For example, fiber-reinforced composites are developed to replace aluminum alloys to provide high strength and fairly high stiffness at low weight without corrosion and fatigue.
For predicting the structural behaviors of such composite materials, computer aided engineering analysis has been used. However, numerically simulating embedded bi-materials using prior art approaches have a number of shortcomings.
A simplified embedded bi-material model in two-dimension and a corresponding prior art FEM model 120 are shown in FIG. 1. The bi-material 100 contains two materials: outer (base) material 102 and inner (embedded or immersed) material 104. The conventional FEM for numerically simulating such material requires a matching or conforming mesh across the material interface 110. Standard finite element shape functions are used to approximate the solution of the underlying boundary value problems. However, generating matching meshes (e.g., FEM mesh 120) suitable for the FEM is difficult in particular having interfaces in irregular geometries. Most of time, construction of matching meshes in interface problem requires substantial user interaction and is time-consuming. One example of the difficulty is shown as irregular grids in a region 122 surrounding the inner material.
Other prior art approaches have problems also. In one example, motar FEM uses a domain decomposition technique to treat mismatching meshes. However, motar FEM requires using Lagrange multipliers that sometimes violate the so-called “inf-sup” condition, which can lead to numerical instability (i.e., no solution). In another examples, “generalized FEM (GFEM)”, “the extended FEM (XFEM)”, “immersed FEM (IFEM)” have been used but the results are too expensive to achieve (e.g., very high computational requirements/costs or not clear to apply in three-dimension).
Another prior art approach is based on meshfree method. Meshfree method has become one of the focused research topics during the 1990's. Many applications of using mesh-free analysis have been achieved in the past decade.
An exemplary mesh-free model 200 is shown in FIG. 2. A physical domain Ω 202 and its boundary or border Γ 203 are depicted. To represent the physical domain 202, a plurality of meshfree nodes 204 are used. The meshfree nodes representing the physical domain do not have a particular pattern. They may be regularly spaced or in arbitrary locations within the domain 202. These meshfree nodes may be located in the interior or on the boundary of the physical domain. Each of the nodes 204 contains a domain of influence or support 206-208. Terms “domain of influence” and “support” are used interchangeably hereinafter. The size and shape of the support for each node are arbitrary. For example, the shape of the support can be quadrilateral 206 or circular 208. In the case of three-dimensional support, the shape of the support may be spherical. The size of the support can be a one square foot or a 16-in radius circle support. The support can have irregular geometric shape.
Due to the flexibility of the meshfree nodal representation 204 of the physical domain 202, a practical way to create a computer model for the meshfree method is to use the FEM model's nodal data that is readily generated from a pre-processing software package. The pre-processing software may be a stand-alone software package or a built-in portion of an engineering design or analysis computer program product package.
However, for simulating structural behaviors of an embedded bi-material, prior art meshfree methods require adding interface constraints and a set of interface nodes to ensure the visibility of the interface in the numerical simulation results. Adding the interface nodes must be done manually. Further, each added node's integration cell must also be adjusted such that the domain integration can be properly conducted. Extending the manual nodal adjustment in the three-dimensional cases, it is not only nontrivial but sometimes impossible.
It would, therefore, be desirable to have a new improved computer aided engineering analysis method and system for numerically simulation structural behaviors of embedded bi-materials.